Electrical coils for generating magnetic fields

ABSTRACT

A novel approach to the mathematical analysis of the magnetic field produced by the coil windings of accelerator beam bending/focussing magnets has led to the proposal for inserting accurately located and dimensioned azimuthal spaces in the windings. The spaces provide a simple means for improving the uniformity of the magnetic field. A mathematically specified pattern of end winding reduces their disturbing influence.

an in 1 31 7v. m m m m t H e t a P S t 3 Al S dd mm C S T N E mm .mS m nN U M U S D L E I F Rm OT m S m mm LG AN CI T A R E N E EG M U [75]Inventor: John Herbert Coupland, High Hol- 3,423,706 1/1969 Sampson eta1...................335/213 X burn, Abingdon, England 3,461,410 8/1969Beth 3,483,493 12/1969 Kafka..................,L....:::........335/2l6[73] Assignee: Science Research Council, London,

England Primary ExaminerGeorge Harris [22] Filed:

Sept 11, 1970 AttorneyLars0n, Taylor and Hinds [57] ABSTRACT A novelapproach to the mathematical analysis of the magnetic field produced bythe coil windings of ac- 211 Appl. No.: 71,590

Foreign Application Priority Data Sept. 18, 1969 GreatBritain................... beam bendmg/mcussmg magnets has led theproposal for inserting accurately located and dimensioned azimuthalspaces in the windings. The

spaces provide a simple means for improving the uniformity of themagnetic field. A mathematically specified pattern of end windingreduces their disturbing influence.

60 m M5 81 31 0 3 l 2 m 3 C QML Um UN 55 [58] Field ofSearch.......................335/2l0, 213, 216

10 Claims, 6 Drawing Figures ELECTRICAL COILS FOR GENERATING MAGNETICFIELDS BACKGROUND OF THE INVENTION The invention relates to electricalcoils for generating magnetic fields and more particularly to such coilsfor generating the magnetic fields for beam bending or beam focusing incharged particle accelerators.

If one considers long electrical conductors arranged around the surfaceof an infinitely long cylinder, with the lengths of the conductorsparallel to the axis of the cylinder, then with axial current flow inthe conductors and a cos 0 distribution of current density around thesurface of the cylinder, a uniform magnetic dipole field within thecylinder will be generated. A cos 20 distribution of current densitywill generate a pure magnetic quadrupole field within the cylinder.

Electrical coils for generating magnetic fields for beam bending or beamfocusing in charged particle accelerators are constructed so as toapproximate as closely as manufacturing difficulties will allow to theideal situation described above. It will be appreciated that thecylinder cannot be infinitely long and, in practice, the windings are inthe form of a coil, a winding running up the length of one side of thecylinder crossing over at an end to run back down the other side.

Although it is not necessary in the calculation to limit the winding toa thin shell, the winding thickness in practice is likely to berelatively small if a high field superconductor is used. A high currentdensity winding is particularly important for a multipole magnet inorder to limit the volume of conductor required.

One proposal for approximating the sinusoidal current densitydistribution is made by W. B. Sampson in Proceedings of theInternational Conference on Magnet Technology, Oxford, 1967, page 574.In this proposal, the circumference of the cylinder is divided into 10equi-depths slots in which are wound the appropriate average number ofconductors for a stepwise approximation to the sine wave.

Another, constructionally simpler, proposal is to use a constant currentdensity winding, a constant radial or slot depth, but choose theazimuthal length of the windings so as to reduce field errors in thebeam space. Such a proposal is adopted by Asner and Iselin inProceedings of the International Conference on Magnet Technology,Oxford, 1967, page 32, who propose further refinement by the addition offurther coils wound radially outside the basic coil.

The present invention is based upon a novel approach in the mathematicalanalysis of the magnetic field generated by electric current in thewindings from which it has been appreciated that considerableimprovement in the field uniformity (or field purity in the case ofmagnetic quadrupole fields or fields of higher orders) may be achievedby the insertion of accurately located and dimensioned spacers withinthe coil windings.

SUMMARY OF THE INVENTION The invention provides, in one of its aspects,an electrical coil for generating a magnetic field, of order N whichcoil is of the form in which the windings lie in a bundle between twoparallel planes spaced apart, the windings lying in two side runs andtwo end runs where the windings cross from one side run to the other,the general direction of the lengths of the windings in the two sideruns being parallel to the said two planes, the side runs beingsubstantially longer than the end runs so that the magnetic field isprincipally defined by the side runs, the windings being arranged sothat there is at least one region, but not more than 4N regions, eachregion being defined in cross-section by an area within each perimeterrespectively of the bundle of windings as seen in cross-section in thetwo side runs, from which region the windings are absent, the number ofwindings per unit cross-sectional area in the side runs beingsubstantially constant throughout the regions where windings arepresent, and the locationand extent of the region or regions where thewindings are absent being selected to enhance the uniformity or purityof the magnetic field generated by the coil.

It is an important feature of the invention that the windings in thesaid two side runs are straight and parallel with one another.

The invention provides, in another of its aspects, an electrical coilfor generating a magnetic field wherein the windings of the coil, asseen in cross-section, are distributed within boundaries defined byconcentric circles and generally radially extending lines at theazimuthal limits of the coil windings, and wherein the windings are soarranged that there is at least one region defined by an area within thesaid boundaries as seen in cross-section, from which region the windingsof the coil are absent, the location and extent of the region beingselected to enhance the uniformity or purity of the magnetic fieldgenerated by the coil.

Preferably said area defining the region comprises a sector of the outercircular boundary truncated by the inner circular boundary.

Preferably the coil windings are divided by azimuthal spacers insertedin the windings, the spacers occupying the said regions as definedabove.

For generating a uniform dipole magnetic field, the arrangement maycomprise coil windings which, as seen in cross-section, are withinboundaries defined by concentric circles, the azimuthal limits of thecoil windings being at 67.40 in each quadrant measured from a referenceaxis, and spacers in each quadrant, the azimuthal extent of the spacersbeing from 43.50 to 52.60 measured from the reference axis, the coilwindings being otherwise uniformly distributed within the boundaries.

For generating a quadrupole field of high purity, the azimuthal limitsof the coil windings, of which limits there will be two in each quadrantdefining the four magnetic pole regions, and the azimuthal extents ofthe spacers, of which there will be two in each quadrant, may be derivedby dividing by two the aforementioned angles for the dipole field coil.Similarly, the corresponding angles for a sextupole field of high puritymay be derived by dividing by three the aforementioned angles for thedipole field coil, and so on for higher order fields.

While the fields generated by coils with limits so defined will be ofimproved uniformity or purity as compared with the fields produced bysimilar coils without spacers, the dipole field may be still furtherimproved if the azimuthal limits of the coil windings are at 7l.8l ineach quadrant measured from the reference axis with spacers from 33.3 8to 37.l7 and from 53.l6 to 63.36".

Corresponding improvement in higher order fields may be obtained byincluding additional spacers, but the higher the number of spacers, themore stringent become the manufacturing tolerances on dimensions andlocations of the spacers which have to be met. In practice, one spacerper pole for quadrupole fields and fields of higher orders, and twospacers per pole for dipole fields, is a likely application of thetechnique of insertion of spacers in windings of magnets foraccelerators.

BRIEF DESCRIPTION OF THE DRAWINGS Specific constructions of electricalcoil embodying the invention will now be described by way of example and.with reference to the accompanying drawings in which:

FIGS. 1 to 3 are cross-sectional views of three forms of coil,

FIG. 4 is a perspective view of a winding for a form of coil,

FIG. 5 is a development showing a pattern of end winding of a coil, and

FIG. 6 is a section on line B-B of FIG. 5.

DESCRIPTION OF PREFERRED EMBODIMENTS FIG. I is included for illustratingthe principles of the mathematical analysis on which the presentinvention is based and shows windings of the form adopted by Asner andlselin as mentioned above for generating a quadrupole field.

In the FIGS. 1 to 3 the general form of the magnetic field is indicatedby arrowed lines.

Referring to FIG. 11, there are four coil windings, one in eachquadrant. The reference R and associated arrows shows the extent of thebundle of windings 11 in one run of one of the coils. At the ends, thewindings l I cross over into the other run, the extent of which isindicated by the reference S. The perimeter or boundary of the bundle ofwindings in the run R, as seen in crosssection, is defined by an outercircle of radius r,,, an inner circle of radius r and radial lines atazimuthal locations defined by angle 0 equal to 60 and 90. The run S hasits boundaries defined by the same circles, but the azimuthal limits areat 0 0' and 0 30. The other three coils are symmetrically arranged andthe azimuthal extent of the coils is defined by the angle indicated as2q5, where, in this case, 4: 30. The four poles of the magneticquadrupole field generated by passing electrical currents through thewindings 11 in the senses indicated by the dots and crosses in thewindings are located in the four pole gaps encompassed by the windingsin each quadrant. Measured from the reference axis BOA, these pole gapsextend from 30 to 60 in each quadrant.

It will be seen that the windings of the coil in the first quadrant inFIG. 1 lie in a bundle between two parallel planes spaced apart. Theplanes extend perpendicularly to the plane of the paper and contain thetwo lines respectively marked 12 and 13. The FIG. 1 shows incross-section the side runs R and S which extend perpendicularly to theplane of the paper a distance which is substantially greater than thelength of the end runs where the windings cross over from side run S toside run R. In a typical example for a magnet for an accelerator theratio of the length of the side runs to the length of the end runs maybe between :1 and 10: 1.

For the mathematical analysis on which the present invention is based,at any radius r, the current density j in a shell of thickness dr isFourier analyzed into its angular components j cosnli, from which thevector potentials A, at a point (r, 0) may be calculated for pointsinside and outside the current shell using the formulas:

The origin of 0 is chosen to exclude terms sin n6, and in the case oftwo-fold symmetry as is required for a quadrupole, n will take only thevalues 2, 6, 10, 14, or (4p 2) where p is an integer. If the Fouriercoefficients j are independent of radius as in the case of sectoredcoils as shown in FIG. 1, then simple integration of these radialexpressions gives by superposition the combined effect for a thick coilof inner and outer radii r and r,,. At points in the winding, A is givenby the following expressions, whence by the usual curl relations thecomponents of field within the coil are obtained. These are necessary inconsidering the mechanical forces on the conductors and in the case ofsuperconductors also in checking that the critical field is not exceededfor the current density in question.

Hence for A 1 g 7 -4 n+2 r 1 r cos n6 n2 b T where and which is aspecial case. Each of these expressions must of course satisfy thePoisson like equation for the region r to r,,, that is where n is thepermittivity of free space.

For equal and equispaced coils each of angular length 20, as in FIG. I,the Fourier coefficients of the current density are obtained as wherethe total azimuthal field within the aperture will be given by 4 [cos20+C cos 60 By performing the analysis this way the amplitudes of theoctupole and higher harmonics are immediately to hand in convenient formfor trajectory calculations.

By way of illustration component values are given in Table I for coilsof configuration as shown in FIG. 1 in which the azimuthal half-lengthof the coils is (b 30. The values of C, are estimated for coils with r,,r y? and will be a little lower for thicker coils.

In practice the maximum usable aperture will probably be only 1.8 to 1.9r,, due to an inner wall thickness for the windings, or even less ifthere has to be an annular space for a temperature transition 300 K to4.2 K. The maximum harmonic error at 0.8 r, of 1.2 percent due to C maybe acceptable for a number of applications.

However, an order of magnitude improvement may be obtained by increasingthe length of each coil and introducing a simple azimuthal space in thewinding of each coil, in effect replacing each single coil with twocoils.

This is illustrated in FIG. 2, where the spaces in the windings areformed by spacers 14. These spacers 14 extend along the length of theruns of each coil and it can be seen that their general location isdefined, in cross-section, by an area within each perimeter respectivelyof the bundle of windings 11 as seen in cross-section in the two sideruns of each coil.

The parameters computed for the coil are an overall azimuthalhalf-length of 5 33.70 and an azimuthal space from 2l.75 to 26.30. Thecomponent values, corresponding to those given in Table I, for thiscoil, again for r, r VTare given in the following I]:

Within an aperture of 1.6 r,, the field error from any of the harmonicsis now less than 0.1 percent. On the other hand, the manufacturingtolerance on the angular boundaries is correspondingly tighter. Forexample, a change of less than 0.05 in any of the boundaries is enoughto reduce j,, to O.

In principle, better fields result from introducing further carefullychosen spaces in the windings, but the windings have to be moreaccurately defined, with their boundaries accurate to 0.01".

Apart from the different radial factors affecting the effectivenesswithin the useful aperture of the different current harmonics and hencethe preferred compromise, the same coil proportions will be best for auniform dipole field or for a sextupole magnet after scaling the anglesby 2 or respectively. Whilst the quality of the sextupole field will beimproved and is likely to be more than adequate, the quality of thedipole field will be correspondingly poorer due to the lower orderattenuation of the harmonic terms.

For generating a dipole field of high uniformity, it is considered to beworthwhile including two spacers in the coil winding for each pole i.e.effectively using three coils per pole. Such an arrangement is shown inFIG. 3. One side run 15 of the coil windings for forming pole 16 has anazimuthal extent from 6 0 to 0 7 1 .84. The other side run 17 issymmetrically arranged in the second quadrant. The other coil windinghaving runs 18 and 19 for forming pole 21 is similarly symmetricallyarranged in the third and fourth quadrants.

Each run 15, 17, 18, 19 has an azimuthal spacer 22 from 33.33 to 37.12as measured from the reference axis COD. Each run l5, 17, 18, 19 alsohas an azimuthal spacer 23 from 53.14 to 63.38 measured from thereference axis COD.

The following'table III shows the harmonic components for the coilarranGement of FIG. 3:

All the lower order components, C to C are approximately zero and inprinciple may be made exactly zero by defining the five angularboundaries to a greater precision. The most significant errors are dueto the components C and C where the impure dipole field as characterizedby the azimuthal component is given by:

I +62%?) cos 50 +C T-) oosn 0 whore At 70 percent of full aperture, thecontribution from (3, will be 0.025 percent and that from C will be0.013 percent. The components C and C could also be made zero by theintroduction of a further azimuthal space in the windings, but inpractice it is believed that the field quality would be limited bYfailure in constructionto realize the strict symmetry implicit in theanalysis. Further, if a superconductor is used, diamagnetic effects inthe windings will spoil the field quality if a wire conductor is notused.

FIG. 4 is a perspective view of one layer of windings for one half of acoil for a dipole field. Side runs corresponding to 15 and 17 of FIG. 3can be seen, as indicated by references 15a and 17a in FIG. 4. Anazimuthal space at 23a can also be seen. The figure is principallyincluded, however, to illustrate the general form of the end windings 31and 32.

the integral being taken along the beam paths through the magnet andparallel to the z axis.

With the arrangement as shown in FIG. 4, the end windings 31, 32 make nocontribution to the above integral. This arrangement is ,therefore asatisfactory solution to the end winding problem where space permits theradial extension at the ends.

However, mathematical analysis has shown that a satisfactory approach tothe above condition may be achieved, without the large radial extension,by following an end winding pattern illustrated in FIGS. 5 and 6.

The cross-sectional view of FIG. 6 shows windings (the sectioned areas)similar to FIG. 3 for a dipole field, but with only one azimuthal spacein each quadrant. FIG. 5 is a development for showing in plan a typicalimproved end winding pattern, the figure showing only the windings inthe quadrant marked A of FIG. 6. For the development, the windings areuncurled so that their curvature from 0 to 0 90 apparent in FIG. 6'isshown planar in FIG. 5. I

As indicated'on the drawing, FIG. shows part of the axial winding, B--Bmarking the section line of FIG. 6, and one quadrant of end winding. Theplan shows a scale of 0 from 0 to 90 for a quadrant of a dipole magnet(the scale would be 0 from 0 to 45 for a quadrupole magnet, etc.) andthe other axis corresponds with the axial distance 2. The conductors 33,34, 35, 36, 37, 38 are arranged parallel to each other and having aninclination 4) to the z axis.

To satisfy the requirement, mentioned above, that is constant, the axialcomponent of the nett current element in the end winding configurationhas to show a similar Fourier analysis with angle 0 as for the sidewindings. To satisfy this condition for an ideal pure c050 system, asolution is:

A 0cot' +L(0)=Bcos0 where A and B are constants and L(0) is the axiallength as shown (see FIG. 5) for conductor 38 only.

Suitable approximate values of iii are indicated in FIG. 5 for thevarious ranges of 0. However, does not need to be constant, or even thesame at each 0 for all of the inclined conductors. If is not the sameat'each 0 for all the conductors, the pattern will necessarily be morecomplex.

The concentric type of construction described with reference to theaccompanying drawings lends itself to the making of a compact combinedfunction magnet, for example where an inner section comprises aquadrupole winding with two or possibly three coils per pole and anouter section comprises a dipole winding having three coils per pole. Inthis way, a beam of charged particles, for example, may be acted uponsimultaneously by a dipole and a quadrupole field.

The self-magnetic field in the outer windings may be derived bydifferentiating the solutions, equation (1) above, and adding thevarious harmonic components. To this must be added terms of the type r""cosn0 due to the decaying quadrupole field of the inner section. For theinner windings, the net field will be given in a similar way fromequation (2) above plus the uniform field from the dipole winding. Aseparate treatment has to be applied to the end sections of the coilwhere field increases can also occur.

The choice of the dipole windings section as the outer section ishelpful in improving the quality of the dipole field in the usefulaperture and at the same time minimizing the dimensions of thequadrupole, so saving in winding material, especially as the fieldgradient B 9 r varies slowly as In (r,,/r,,). With the dipole winding,the magnetic field depends directly upon the thickness of the winding(r, r,,) and consequently there is less incentive to minimize the meanradius. A further advantage of the concentric arrangement is that it iseasier to ensure that the two magnets are self-aligned to a common axis.

The technique described above for improving the uniformity or purity ofthe magnetic field is also advantageous where superconducting windingsare employed because one avoids unnecessary magnetic field rise in thesuperconducting coils themselves.

The invention is not restricted to the details of the foregoingexamples. For instance, the side runs of the coil windings need notnecessarily be confined within concentric circular boundaries as seen incross-section.

That is, the side runs of the windings need not conform to the surfaceof a cylinder. The arrangement may, for example, be such that, as seenin cross-section, the side runs of the windings are contained withinrectangular boundaries. In this case, the field uniformity or purity cansimilarly be enhanced by spacers of predetermined dimensions andlocations. However, the mathematical calculation of the parameters ofthe spacers is considerably more complex than that for theabovedescribed concentric arrangements.

Further, the windings may be formed from wide, thin tapes ofsuperconductor, which may be braided.

Iclaim:

1. An electrical coil for generating a magnetic field of order N, whichcoil .is of the form in which the windings lie in a bundle'between twoparallel planes spaced apart, the windings lying in two side runs andtwo end runs where the windingscross from one side run to the other, thegeneral direction of the lengths of the windings in the two side runsbeing parallel to the said two planes, the side runs being substantiallylonger than the end runs so that the magnetic field is principallydefined by the side runs, the windings being arranged so that there isat least one region, but not more than 4N regions, each region beingdefined in crosssection by an area within each perimeter respectively ofthe bundle of windings as seen in cross-section in the two side runs,from which region the windings are absent, the number of windings perunit cross-sectional area in the side runs being substantially constantthroughout the regions where the windings are present, and the locationand extent of the region or regions where the windings are absent beingselected to enhance the uniformity or purity of the magnetic fieldgenerated by the coil.

2. An electrical coil as claimed in claim 1, wherein the windings in thesaid two side runs are straight and parallel with one another.

3. An electrical coil for generating a magnetic field as claimed inclaim 2 wherein the windings of the coil, as seen in cross-section, aredistributed within boundaries defined by concentric circles andgenerally radially extending lines at the azimuthal limits of the coilwindings, and wherein the said area defining the region from which thewindings are absent is an area within the said boundaries as seen incross-section.

4. An electrical coil as claimed in claim 3, wherein the said areadefining the region comprises a sector of the outer circular boundarytruncated by the inner circular boundary.

5. An electrical coil as claimed in claim 3, wherein the coil windingsare divided by azimuthal spacers inserted in the windings, the spacersoccupying the said regions as defined above.

6. An electrical coil as claimed in claim 5, wherein, for generating auniform dipole magnetic field, the arrangement comprises coil windingswhich, as seen in cross-section, are within boundaries defined byconcentric circles, the azimuthal limits of the coil windings being at67.40 in each quadrant measured from a reference axis, and spacers ineach quadrant, the azimuthal extent of the spacers being from 43.50 to52.60 measured from the reference axis, the coil windings beingotherwise uniformly distributed within the boundaries.

7. An electrical coil as claimed in claim 5, wherein,

for generating a quadrupole field of high purity, the azimuthal limitsof the coil windings, of which there are two in each quadrant definingfour magnetic pole regions, are at 33.70 in each quadrant measured frommutually perpendicular reference axes, and the azimuthal extents of thespacers, of which there are two in each quadrant, are from 2l.75 to26.30 measured from the reference axes.

8. An electrical coil as claimed in claim 6, wherein, for generating amagnetic field of order N, the azimuthal limits of the coil windings, ofwhich limits there will be N, in each quadrant defining the N magneticpole regions, and the azimuthal extents of the spacers, of which thereare N in each quadrant, are derived by dividing by N the anglesmentioned in claim 6 for the dipole field coil.

9. An electrical coil as claimed in claim 5, wherein, for generating auniform dipole magnetic field, the arrangement comprises coil windingswhich, as seen in cross sec tion, are within boundaries defined byconcentric circles, the azimuthal limits of the coil windings being at71.84 in each quadrant measured from a reference axis, and two spacersin each quadrant, the azimuthal extent of the spacers being from 33.33to 37.l2 and from 53.l4 to 63.38".

10. An electrical coil for generating a magnetic field, which coil is ofthe form in which the windings lie in a bundle between two parallelplanes spaced apart, the windings lying in two side runs and two endruns where the windings cross from one side run to the other, thegeneral direction of the lengths of the windings in the two side runsbeing parallel to the said two planes, the side runs being substantiallylonger than the end runs so that the magnetic field is principallydefined by the side runs, the windings in the end runs being arranged sothat for any length of winding of azimuthal location 6 and inclinationd) to a z axis defined parallel to the said general direction of thelengths of the windings in the side runs, the following condition issatisfied:

A 0cot'+L(0)=Bcos0 where A and B are constants and L(6) is the length ofthe winding continuing in the direction of the side run beyond an endplane, the said end plane being defined as containing the point ofdiversion from side run to end run of the winding for which L(9) iszero.

1. An electrical coil for generating a magnetic field of order N, whichcoil is of the form in which the windings lie in a bundle between twoparallel planes spaced apart, the windings lying in two side runs andtwo end runs where the windings cross from one side run to the other,the general direction of the lengths of the windings in the two sideruns being parallel to the said two planes, the side runs beingsubstantially longer than the end runs so that the magnetic field isprincipally defined by the side runs, the windings being arranged sothat there is at least one region, but not more than 4N regions, eachregion being defined in cross-section by an area within each perimeterrespectively of the bundle of windings as seen in cross-section in thetwo side runs, from which region the windings are absent, the number ofwindings per unit crosssectional area in the side runs beingsubstantially constant throughout the regions where the windings arepresent, and the location and extent of the region or regions where thewindings are absent being selected to enhance the uniformity or purityof the magnetic field generated by the coil.
 2. An electrical coil asclaimed in claim 1, wherein the windings in the said two side runs arestraight and parallel with one another.
 3. An electrical coil forgenerating a magnetic field as claimed in claim 2 wherein the windingsof the coil, as seen in cross-section, are distributed within boundariesdefined by concentric circles and generally radially extending lines atthe azimuthal limits of the coil windings, and wherein the said areadefining the region from which the windings are absent is an area withinthe said boundaries as seen in cross-section.
 4. An electrical coil asclaimed in claim 3, wherein the said area defining the region comprisesa sector of the outer circular boundary truncated by the inner circularboundary.
 5. An electrical coil as claimed in claim 3, wherein the coilwindings are divided by azimuthal spacers inserted in the windings, thespacers occupying the said regions as defined above.
 6. An electricalcoil as claimed in claim 5, wherein, for generating a uniform dipolemagnetic field, the arrangement comprises coil windings which, as seenin cross-section, are within boundaries defined by concentric circles,the azimuthal limits of the coil windings being at 67.40* in eachquadrant measured from a reference axis, and spacers in each quadrant,the azimuthal extent of the spacers being from 43.50* to 52.60* measuredfrom the reference axis, the coil windings being otherwise uniformlydistributed within the boundaries.
 7. An electrical coil as claimed inclaim 5, wherein, for generating a quadrupole field of high purity, theazimuthal limits of the coil windings, of which There are two in eachquadrant defining four magnetic pole regions, are at 33.70* in eachquadrant measured from mutually perpendicular reference axes, and theazimuthal extents of the spacers, of which there are two in eachquadrant, are from 21.75* to 26.30* measured from the reference axes. 8.An electrical coil as claimed in claim 6, wherein, for generating amagnetic field of order N, the azimuthal limits of the coil windings, ofwhich limits there will be N/2 in each quadrant defining the N magneticpole regions, and the azimuthal extents of the spacers, of which thereare N/2 in each quadrant, are derived by dividing by N/2 the anglesmentioned in claim 6 for the dipole field coil.
 9. An electrical coil asclaimed in claim 5, wherein, for generating a uniform dipole magneticfield, the arrangement comprises coil windings which, as seen incross-section, are within boundaries defined by concentric circles, theazimuthal limits of the coil windings being at 71.84* in each quadrantmeasured from a reference axis, and two spacers in each quadrant, theazimuthal extent of the spacers being from 33.33* to 37.12* and from53.14* to 63.38*.
 10. An electrical coil for generating a magneticfield, which coil is of the form in which the windings lie in a bundlebetween two parallel planes spaced apart, the windings lying in two sideruns and two end runs where the windings cross from one side run to theother, the general direction of the lengths of the windings in the twoside runs being parallel to the said two planes, the side runs beingsubstantially longer than the end runs so that the magnetic field isprincipally defined by the side runs, the windings in the end runs beingarranged so that for any length of winding of azimuthal location thetaand inclination phi '' to a z axis defined parallel to the said generaldirection of the lengths of the windings in the side runs, the followingcondition is satisfied: A theta cot phi '' + L( theta ) B cos thetawhere A and B are constants and L( theta ) is the length of the windingcontinuing in the direction of the side run beyond an end plane, thesaid end plane being defined as containing the point of diversion fromside run to end run of the winding for which L( theta ) is zero.